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Math 2R03 (Theory of Linear Algebra) Homework Assignment 3
All of the questions from Part A will be graded. One of the questions from Part B will be graded in detail, while the other will be marked for completion. Assignments will be submitted via Crowdmark. You will be graded on your solution and how well you write your proof.
Part A. [Short Questions; 4pts]
Exercise 1. Let U be a subspace of the finite dimensional vector space V , i.e., U ⊆ V . Prove that if dim U = dim V , then U = V .
Remark. This result is very useful to show two vector spaces are equal. We use this fact throughout the course.
Exercise 2. Fix an integer m ≥ 1, and let D ∈ L(Pm+1(R),Pm(R)) be the linear map given by
D(a0 + a1x + a2x 2 + · · · + am+1x m+1) = a1 + 2a2x + 3a3x 2 + · · · + (m + 1)am+1x m.
Prove that dim Null(D) = 1.
Part B. [Proof Questions; 6pts]
Exercise 3. Let V = R 3 and consider the subspaces U = span((1, 0, 0),(0, 1, 0)) and W = span((2, 1, 0),(0, 0, 1)) in V . Find a basis for U ∩ W. Hint. What is dim(U ∩ W)?
Exercise 4. Suppose that v1, . . . , v2022 ∈ F 2021 are 2022 distinct vectors. Prove that for any vector space W and for any linear map T ∈ L(F 2021, W), the vectors T v1, . . . , T v2022 are linear dependent in W.
Additional Suggested Problems. [Not graded]
Problems 2.B #5, 6, 2.C #11, 14, 15, 3.A #4, 7 1